A circle has a radius of $4$. An arc in this circle has a central angle of $135^\circ$. What is the length of the arc? ${8\pi}$ ${135^\circ}$ $\color{#DF0030}{3\pi}$ ${4}$
Solution: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (4) = 8\pi$ The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360^\circ} = \dfrac{s}{c}$ $\dfrac{135^\circ}{360^\circ} = \dfrac{s}{8\pi}$ $\dfrac{3}{8} = \dfrac{s}{8\pi}$ $\dfrac{3}{8} \times 8\pi = s$ $3\pi = s$